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Full text of "Mathematical And Physical Papers - Iii"

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ON THE CONDUCTION OF HEAT IN CRYSTALS. 223 readily from the theory of molecular radiation. In this theory the extent of molecular radiation is supposed to be very great compared with the mean interval between the molecules of a body, so that the body may be treated as continuous. If E, E' be any two elements of the body, situated sufficiently near one another to render their mutual influence sensible, it is supposed that, during the time dt, a quantity of heat proportional to Edt radiates in all directions from E, whereof Er absorbs a portion proportional to E'. On the other hand, E' emits and E absorbs a quantity also proportional, so far as regards only the magnitudes of E, Ef, and dt, to EE'dt. The exchange of heat between E and E' may therefore be expressed by qEE'dt. The quantity q is supposed to be proportional, so far as regards its dependence on the temperatures, to the small difference between the temperatures of E and E'. It will also depend upon the nature of the body, upon the distance EE, and in the case of a crystalline body upon the direction of the line EE'; but we need not now consider its dependence upon these quantities. If the length EE' = s, and if we suppose the extent of internal radiation to be very small, we may express the difference between, the temperatures of E and E by du/ds.s. It follows then from the theory we are considering, that the total flux of heat arises from the exchange of heat between all possible pairs of elements, such as E, E1'; the exchange between any pair E, E' being proportional to the rate of variation of temperature in the direction EE', and accordingly independent of the variation of temperature in other directions. Now suppose the body referred to rectangular axes, and let P be the mathematical point whose coordinates are %, y, z. Conceive the body divided into an infinite number of infinitely small equal elements. Let E be the element which contains P, E' any element in the neighbourhood of P, and consider the partial flux in the neighbourhood of P which arises from the exchange of heat between all pairs of elements which have the same relative position as E and E'. Through P draw an elementary plane S, which it will be convenient to consider as infinitely large compared with the dimensions of the elements such as E, and conceive S to assume in succession all possible directions by turning round P. The partial flux across S will vary as the number of